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Theorem relcmpcmet 23848
Description: If 𝐷 is a metric space such that all the balls of some fixed size are relatively compact, then 𝐷 is complete. (Contributed by Mario Carneiro, 15-Oct-2015.)
Hypotheses
Ref Expression
relcmpcmet.1 𝐽 = (MetOpen‘𝐷)
relcmpcmet.2 (𝜑𝐷 ∈ (Met‘𝑋))
relcmpcmet.3 (𝜑𝑅 ∈ ℝ+)
relcmpcmet.4 ((𝜑𝑥𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
Assertion
Ref Expression
relcmpcmet (𝜑𝐷 ∈ (CMet‘𝑋))
Distinct variable groups:   𝑥,𝐷   𝑥,𝐽   𝜑,𝑥   𝑥,𝑅   𝑥,𝑋

Proof of Theorem relcmpcmet
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 relcmpcmet.2 . 2 (𝜑𝐷 ∈ (Met‘𝑋))
2 metxmet 22871 . . . . . . 7 (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋))
31, 2syl 17 . . . . . 6 (𝜑𝐷 ∈ (∞Met‘𝑋))
43adantr 481 . . . . 5 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝐷 ∈ (∞Met‘𝑋))
5 simpr 485 . . . . 5 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (CauFil‘𝐷))
6 relcmpcmet.3 . . . . . 6 (𝜑𝑅 ∈ ℝ+)
76adantr 481 . . . . 5 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝑅 ∈ ℝ+)
8 cfil3i 23799 . . . . 5 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷) ∧ 𝑅 ∈ ℝ+) → ∃𝑥𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
94, 5, 7, 8syl3anc 1363 . . . 4 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → ∃𝑥𝑋 (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
103ad2antrr 722 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐷 ∈ (∞Met‘𝑋))
11 relcmpcmet.1 . . . . . . . . 9 𝐽 = (MetOpen‘𝐷)
1211mopntopon 22976 . . . . . . . 8 (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋))
1310, 12syl 17 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ (TopOn‘𝑋))
14 cfilfil 23797 . . . . . . . . 9 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
153, 14sylan 580 . . . . . . . 8 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → 𝑓 ∈ (Fil‘𝑋))
1615adantr 481 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (Fil‘𝑋))
17 simprr 769 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)
18 topontop 21449 . . . . . . . . . . 11 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1913, 18syl 17 . . . . . . . . . 10 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝐽 ∈ Top)
20 simprl 767 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑥𝑋)
216rpxrd 12420 . . . . . . . . . . . . 13 (𝜑𝑅 ∈ ℝ*)
2221ad2antrr 722 . . . . . . . . . . . 12 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑅 ∈ ℝ*)
23 blssm 22955 . . . . . . . . . . . 12 ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑅 ∈ ℝ*) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
2410, 20, 22, 23syl3anc 1363 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝑋)
25 toponuni 21450 . . . . . . . . . . . 12 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
2613, 25syl 17 . . . . . . . . . . 11 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑋 = 𝐽)
2724, 26sseqtrd 4004 . . . . . . . . . 10 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ 𝐽)
28 eqid 2818 . . . . . . . . . . 11 𝐽 = 𝐽
2928clsss3 21595 . . . . . . . . . 10 ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ 𝐽) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝐽)
3019, 27, 29syl2anc 584 . . . . . . . . 9 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝐽)
3130, 26sseqtrrd 4005 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋)
3228sscls 21592 . . . . . . . . 9 ((𝐽 ∈ Top ∧ (𝑥(ball‘𝐷)𝑅) ⊆ 𝐽) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
3319, 27, 32syl2anc 584 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
34 filss 22389 . . . . . . . 8 ((𝑓 ∈ (Fil‘𝑋) ∧ ((𝑥(ball‘𝐷)𝑅) ∈ 𝑓 ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ⊆ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
3516, 17, 31, 33, 34syl13anc 1364 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓)
36 fclsrest 22560 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
3713, 16, 35, 36syl3anc 1363 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) = ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
38 inss1 4202 . . . . . . 7 ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fClus 𝑓)
39 eqid 2818 . . . . . . . . 9 dom dom 𝐷 = dom dom 𝐷
4011, 39cfilfcls 23804 . . . . . . . 8 (𝑓 ∈ (CauFil‘𝐷) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
4140ad2antlr 723 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fClus 𝑓) = (𝐽 fLim 𝑓))
4238, 41sseqtrid 4016 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽 fClus 𝑓) ∩ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ⊆ (𝐽 fLim 𝑓))
4337, 42eqsstrd 4002 . . . . 5 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓))
44 relcmpcmet.4 . . . . . . 7 ((𝜑𝑥𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
4544ad2ant2r 743 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp)
46 filfbas 22384 . . . . . . . . . 10 (𝑓 ∈ (Fil‘𝑋) → 𝑓 ∈ (fBas‘𝑋))
4716, 46syl 17 . . . . . . . . 9 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → 𝑓 ∈ (fBas‘𝑋))
48 fbncp 22375 . . . . . . . . 9 ((𝑓 ∈ (fBas‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ∈ 𝑓) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
4947, 35, 48syl2anc 584 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓)
50 trfil3 22424 . . . . . . . . 9 ((𝑓 ∈ (Fil‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → ((𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
5116, 31, 50syl2anc 584 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ↔ ¬ (𝑋 ∖ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ 𝑓))
5249, 51mpbird 258 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
53 resttopon 21697 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) ⊆ 𝑋) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
5413, 31, 53syl2anc 584 . . . . . . . . 9 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
55 toponuni 21450 . . . . . . . . 9 ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (TopOn‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
5654, 55syl 17 . . . . . . . 8 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)) = (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))
5756fveq2d 6667 . . . . . . 7 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (Fil‘((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (Fil‘ (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
5852, 57eleqtrd 2912 . . . . . 6 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘ (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))))
59 eqid 2818 . . . . . . 7 (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) = (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))
6059fclscmpi 22565 . . . . . 6 (((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ Comp ∧ (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) ∈ (Fil‘ (𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))))) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
6145, 58, 60syl2anc 584 . . . . 5 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅)
62 ssn0 4351 . . . . 5 ((((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ⊆ (𝐽 fLim 𝑓) ∧ ((𝐽t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅))) fClus (𝑓t ((cls‘𝐽)‘(𝑥(ball‘𝐷)𝑅)))) ≠ ∅) → (𝐽 fLim 𝑓) ≠ ∅)
6343, 61, 62syl2anc 584 . . . 4 (((𝜑𝑓 ∈ (CauFil‘𝐷)) ∧ (𝑥𝑋 ∧ (𝑥(ball‘𝐷)𝑅) ∈ 𝑓)) → (𝐽 fLim 𝑓) ≠ ∅)
649, 63rexlimddv 3288 . . 3 ((𝜑𝑓 ∈ (CauFil‘𝐷)) → (𝐽 fLim 𝑓) ≠ ∅)
6564ralrimiva 3179 . 2 (𝜑 → ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅)
6611iscmet 23814 . 2 (𝐷 ∈ (CMet‘𝑋) ↔ (𝐷 ∈ (Met‘𝑋) ∧ ∀𝑓 ∈ (CauFil‘𝐷)(𝐽 fLim 𝑓) ≠ ∅))
671, 65, 66sylanbrc 583 1 (𝜑𝐷 ∈ (CMet‘𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  cdif 3930  cin 3932  wss 3933  c0 4288   cuni 4830  dom cdm 5548  cfv 6348  (class class class)co 7145  *cxr 10662  +crp 12377  t crest 16682  ∞Metcxmet 20458  Metcmet 20459  ballcbl 20460  fBascfbas 20461  MetOpencmopn 20463  Topctop 21429  TopOnctopon 21446  clsccl 21554  Compccmp 21922  Filcfil 22381   fLim cflim 22470   fClus cfcls 22472  CauFilccfil 23782  CMetccmet 23784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-2o 8092  df-oadd 8095  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ico 12732  df-rest 16684  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-top 21430  df-topon 21447  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-cmp 21923  df-fil 22382  df-flim 22475  df-fcls 22477  df-cfil 23785  df-cmet 23787
This theorem is referenced by:  cmpcmet  23849  cncmet  23852
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